A method and system for solving engineering problems based on a constitutive model driver

Abstract

The application relates to the technical field of data processing, and provides an engineering problem solving method and system based on a constitutive model drive, which comprises the following steps: performing physical item mining on prior knowledge to obtain a candidate constitutive item set; performing data enhancement on experimental data through a Gaussian noise function to obtain an expanded training data set; performing nested optimization on the candidate constitutive item set to obtain an optimal constitutive equation structure and parameter combination; performing explainability verification on the optimal constitutive equation structure and the parameter combination according to a multi-target screening criterion, and establishing an obtained engineering problem constitutive model; and solving a to-be-solved engineering problem through the engineering problem constitutive model to obtain a solving result. The application proposes a construction method of an engineering problem high-precision constitutive model, realizes automatic construction of an explainable engineering constitutive model, and improves a prediction accuracy.

Description

Technical Field

[0001] This invention belongs to the field of data processing technology, and in particular relates to a constitutive model-driven method and system for solving engineering problems. Background Technology

[0002] Current data-driven AI models, such as machine learning and large language models, have demonstrated strong potential in solving engineering problems. These models construct complex nonlinear mapping relationships through massive amounts of data, enabling them to handle multivariable and multi-constraint engineering optimization problems that are difficult to solve using traditional numerical methods. In fields such as materials science, structural engineering, and heat conduction analysis, prediction models based on deep neural networks have been widely applied in performance prediction, parameter optimization, and simulation calculations. Meanwhile, to enhance the physical plausibility of these models, researchers have proposed improved methods that incorporate physical laws, such as physical information neural networks, attempting to introduce physical constraints while maintaining high accuracy. These methods, by embedding partial differential equations or physical laws into the loss function, alleviate the limitations of purely data-driven models to some extent.

[0003] However, existing technical solutions still have significant shortcomings: First, the black-box nature of traditional AI models makes the decision-making logic completely untraceable, and engineers cannot understand the model's internal reasoning process, which seriously hinders the deepening of the understanding of the discipline's mechanisms and the improvement of the model's credibility; Second, existing physical information fusion methods rely excessively on explicit physical formulas and forced constraint optimization paths, and in complex engineering scenarios lacking a complete theoretical foundation or where the physical mechanisms are not yet fully clear, these methods are difficult to achieve high-precision modeling; Third, when dealing with complex engineering problems such as multi-physics coupling and cross-scale effects, current methods often require a large amount of prior knowledge and manual parameter tuning, resulting in low automation and difficulty in building reliable prediction models under limited experimental data conditions. Summary of the Invention

[0004] In view of the shortcomings of the prior art, the purpose of the invention is to provide a constitutive model-driven method and system for solving engineering problems.

[0005] The first aspect of this invention provides a constitutive model-driven method for solving engineering problems, comprising: S1: Perform physical term mining on prior knowledge to obtain a set of candidate constitutive terms; S2: Data augmentation of the experimental data is performed using a Gaussian noise function to obtain an expanded training dataset; S3: Perform nested optimization on the candidate constitutive term set to obtain the optimal constitutive equation structure and parameter combination; S4: Verify the interpretability of the optimal constitutive equation structure and the parameter combination, and establish a constitutive model for the engineering problem based on the verified optimal constitutive equation structure and parameter combination; S5: Solve the engineering problem to be solved using the constitutive model of the engineering problem to obtain the solution result.

[0006] According to the constitutive model-driven method for solving engineering problems provided by the present invention, step S1 further includes: S11: Determine the relevant variable parameters based on the target quantity of the engineering problem; S12: Combine relevant variables into local physics terms; S13: Add exponential functions to multiple local physical terms to eliminate dimensions and obtain a standardized set of candidate constitutive terms.

[0007] According to the constitutive model-driven method for solving engineering problems provided by the present invention, the set of candidate constitutive terms in step S1 is selected based on the physical mechanism of the engineering domain corresponding to the engineering problem to be solved.

[0008] According to the constitutive model-driven method for solving engineering problems provided by the present invention, step S3 further includes: S31: Using the symbolic regression algorithm, perform an outer structure search on the candidate constitutive term set to obtain a candidate formula structure set; S32: By using the differential optimization algorithm, the inner layer parameters of multiple formulas in the candidate formula structure set are optimized to obtain the optimal parameter configuration of multiple formulas; S33: The optimal parameter configuration is evaluated based on the cross-entropy loss function and regularization constraints to obtain the optimal constitutive equation structure and parameter combination.

[0009] According to the constitutive model-driven method for solving engineering problems provided by the present invention, the differential optimization algorithm in step S32 adopts the best1bin strategy.

[0010] According to the constitutive model-driven method for solving engineering problems provided by the present invention, step S4 further includes: S41: Physical constraint verification is performed to check the physical rationality of the structure and parameter combination of the optimal constitutive equation, and the physical constraint verification results are obtained. S42: Based on the accuracy index and the verification results of the physical constraints, evaluate the optimal constitutive equation structure and parameter combination to obtain the constitutive model of the engineering problem.

[0011] According to the constitutive model-driven method for solving engineering problems provided by the present invention, the engineering problem to be solved in step S5 includes: Problems include process parameter optimization, cross-scale performance prediction, and finite element simulation.

[0012] A second aspect of the present invention provides a constitutive model-driven engineering problem-solving system, comprising: Mining module: Used to mine physical terms from prior knowledge to obtain a set of candidate constitutive terms; Augmentation module: Used to augment experimental data using a Gaussian noise function to obtain an expanded training dataset; Optimization module: used to perform nested optimization on the candidate constitutive term set to obtain the optimal constitutive equation structure and parameter combination; Verification module: used to verify the interpretability of the optimal constitutive equation structure and the parameter combination, and to establish a constitutive model of the engineering problem based on the verified optimal constitutive equation structure and parameter combination; The solution module is configured to solve the engineering problem constitutive model established by the verification module and obtain the solution result.

[0013] A third aspect of the present invention provides a constitutive model-driven engineering problem-solving device, comprising: A memory and at least one processor, wherein the memory stores instructions; At least one of the processors invokes the instructions in the memory to cause a constitutive model-driven engineering problem-solving apparatus to execute a constitutive model-driven engineering problem-solving method as described in any of the preceding claims.

[0014] A fourth aspect of the present invention provides a computer-readable storage medium storing instructions that, when executed by a processor, implement a constitutive model-driven engineering problem-solving method as described in any one of the preceding descriptions.

[0015] This invention provides a constitutive model-driven method and system for solving engineering problems. By using a large language model to mine physical terms from relevant literature and prior knowledge of engineering problems, it can systematically extract and integrate physical mechanism information scattered in a large amount of literature. This avoids the limitations of traditional methods that rely on personal experience and subjective judgment, and significantly improves the comprehensiveness and accuracy of physical term identification, providing a richer and more reliable physical foundation for subsequent modeling.

[0016] This invention uses a Gaussian noise function to augment a small amount of experimental data, effectively solving the key problem of scarce experimental data in engineering practice. It expands a small amount of original data into a large number of training samples, which not only significantly reduces experimental costs and time investment, but also fully explores the potential information in the limited data, enabling the model to achieve high-precision prediction results even under small sample conditions. This provides a realistic and feasible solution for resource-constrained engineering projects.

[0017] Secondly, this invention also performs nested optimization processing on the candidate constitutive term set through symbolic regression algorithm and difference optimization algorithm, realizing the synergistic evolution of formula structure search and parameter optimization. The outer symbolic regression algorithm can efficiently search for the optimal structure combination in the huge formula space, while the inner difference optimization algorithm ensures that each candidate formula can find the local optimal parameter configuration. The dual-layer optimization strategy not only improves the global search efficiency, but also ensures that the mathematical expression of the final model is both in line with physical intuition and has the best fitting performance.

[0018] Furthermore, this invention performs performance evaluation on each formula based on the cross-entropy loss function and regularization constraints, effectively balancing the relationship between model complexity and prediction accuracy, preventing overfitting, and ensuring that the model has good generalization ability. The optimal constitutive equation is verified through physical constraints to ensure that the generated mathematical model not only performs well in a statistical sense, but more importantly, maintains rationality and interpretability in a physical sense. This allows engineers to intuitively understand the model's prediction logic and physical mechanism, thereby enhancing the model's credibility and practicality.

[0019] Overall, the constitutive model constructed in this invention can be directly applied to multiple engineering fields such as process parameter optimization, cross-scale performance prediction, and finite element simulation calculation. It provides a high-precision and interpretable prediction tool for complex engineering problems, promotes the transformation of engineering science from experience-driven to theory-guided, and has important scientific value and engineering application prospects. Attached Figure Description

[0020] The accompanying drawings are for illustrative purposes only and are not intended to limit the invention. Throughout the drawings, the same reference numerals denote the same parts. It is obvious that the drawings described below are merely some embodiments of the present invention, and those skilled in the art can obtain other drawings based on these drawings.

[0021] Figure 1 A schematic diagram of the process flow for a constitutive model-driven method for solving engineering problems provided in an embodiment of the present invention; Figure 2 This is a schematic diagram of an apparatus for solving engineering problems based on a constitutive model, as provided in an embodiment of the present invention.

[0022] Figure label: 100. Mining Module; 200. Enhancement Module; 300. Optimization Module; 400. Verification Module; 500. Solving Module. Detailed Implementation

[0023] To enable those skilled in the art to better understand the technical solutions in the embodiments of the present invention, the technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. It should be understood that these descriptions are merely exemplary and are not intended to limit the scope of the present invention. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort should fall within the scope of protection of the present invention.

[0024] Furthermore, descriptions of well-known structures and techniques are omitted in the following description to avoid unnecessarily obscuring the concepts disclosed in this invention.

[0025] In the description of this invention, it should be noted that, unless otherwise explicitly specified and limited, the terms "center," "upper," "lower," "left," "right," "vertical," "horizontal," "inner," and "outer," etc., indicating orientation or positional relationships based on the orientation or positional relationships shown in the accompanying drawings, are only for the convenience of describing the invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation, and therefore should not be construed as a limitation of the invention. Furthermore, the terms "first," "second," and "third" are used for descriptive purposes only and should not be construed as indicating or implying relative importance. The terms "installed," "connected," and "linked" should be interpreted broadly; for example, they can refer to a fixed connection, a detachable connection, or an integral connection; they can refer to a mechanical connection or an electrical connection; they can refer to a direct connection or an indirect connection through an intermediate medium; and they can refer to the internal communication of two components. Those skilled in the art can understand the specific meaning of the above terms in this invention based on the specific circumstances.

[0026] Exemplary embodiments will now be described in detail, examples of which are illustrated in the accompanying drawings. When the following description relates to the drawings, unless otherwise indicated, the same numerals in different drawings denote the same or similar elements. The embodiments described in the following exemplary embodiments do not represent all embodiments consistent with the present invention. Rather, they are merely examples of methods and systems consistent with some aspects of the invention as detailed in the appended claims.

[0027] The embodiments of the present invention are described below with reference to the figures.

[0028] like Figure 1 As shown, the first aspect of the present invention provides a constitutive model-driven method for solving engineering problems, comprising: S1: Perform physical term mining on prior knowledge to obtain a set of candidate constitutive terms.

[0029] Step S1 further includes: S11: Determine the relevant variable parameters based on the objective quantity of the engineering problem; S12: Combine the relevant variables into local physical terms; S13: Add exponential functions to multiple local physical terms to eliminate dimensions and obtain a standardized set of candidate constitutive terms.

[0030] Furthermore, in steps S11 to S13, this invention obtains a set of candidate constitutive terms by mining physical terms from prior knowledge. First, relevant variable parameters are determined based on the target quantity of the engineering problem. That is, key variables affecting the target quantity are identified by analyzing the physical nature and mathematical description of the specific engineering problem. Subsequently, relevant variables are combined into local physical terms through prior experience or interaction with a large language model. If there is no prior experience, semantic analysis and knowledge extraction are performed on massive engineering documents using natural language processing technology based on a large language model to identify the physical relationships between variables and generate physically meaningful combination expressions. Finally, exponential functions are added to multiple local physical terms to eliminate dimensions and obtain a standardized set of candidate constitutive terms. Dimension elimination is achieved by adding power exponent parameters to each physical term to eliminate the dimensional differences between different physical quantities, so that different physical terms can be calculated and compared within the same mathematical framework.

[0031] In step S1, the set of candidate constitutive terms is selected based on the physical mechanism of the engineering field corresponding to the engineering problem to be solved.

[0032] In one embodiment, the set of candidate constitutive terms may include any one or more of the following: input parameter terms, boundary constraint terms, external influence terms, system state terms, intrinsic property terms, and process parameter terms.

[0033] A specific embodiment of the present invention is a welding engineering problem. In the welding engineering problem, the candidate constitutive term set includes a current heat input term, a geometric constraint term, a temperature gradient term, a heat accumulation term, a cooling term, and a welding wire diameter term. The current heat input term characterizes the heat input intensity generated by the electric arc during the welding process and is calculated by a combination of parameters such as welding current, voltage, and welding speed. The geometric constraint term reflects the constraint effect of the weld geometry on heat conduction and stress distribution. The temperature gradient term describes the spatial distribution characteristics of the temperature field during the welding process. The heat accumulation term quantifies the heat accumulation effect of the material during the welding process. The cooling term characterizes the influence of the post-weld cooling process on the material microstructure and properties. The welding wire diameter term reflects the influence of the welding wire geometric parameters on the amount of deposited metal and heat input.

[0034] S2: Data augmentation of the experimental data is performed using a Gaussian noise function to obtain an expanded training dataset.

[0035] Furthermore, in step S2, the present invention uses a Gaussian noise function to augment the experimental data to obtain an expanded training dataset. Specifically, the present invention generates new training samples by superimposing random noise conforming to a Gaussian distribution onto the original experimental data. First, it is necessary to determine the range of noise addition, which is usually set to 5%-15% of the standard deviation of the original data. Then, random perturbation is added to each original data point according to the set noise level to generate multiple variant samples. The Gaussian noise is added in such a way that the variance of the Gaussian process is determined based on the statistical characteristics of the original data and engineering experience. The relationship between data diversity and authenticity is balanced by controlling the noise intensity and the number of generated samples. The expanded training dataset contains the original experimental data and all its noise variant samples, forming a larger training set.

[0036] S3: Perform nested optimization on the candidate constitutive term set to obtain the optimal constitutive equation structure and parameter combination.

[0037] Furthermore, in step S3 of this invention, the optimal constitutive equation structure and parameter combination are obtained by performing nested optimization on the candidate constitutive term set. Nested optimization refers to combining the optimization problems of structure search and parameter optimization at two levels. The outer layer is responsible for searching for the optimal mathematical expression structure, and the inner layer is responsible for finding the optimal parameter configuration for each candidate structure. The two optimizations are nested and co-evolve to find the global optimal solution.

[0038] Step S3 further includes: S31: Using a symbolic regression algorithm, perform an outer structure search on the candidate constitutive term set to obtain a candidate formula structure set.

[0039] Furthermore, in step S31 of this invention, a candidate formula structure set is obtained by searching the outer structure of the candidate constitutive term set using a symbolic regression algorithm. The symbolic regression algorithm is a symbolic expression search method based on a genetic algorithm. Its core idea is to represent mathematical expressions as syntax trees and search for the optimal symbol combination in the expression space through selection, crossover, and mutation operations of the genetic algorithm. Specifically, this invention first initializes a population containing multiple randomly generated mathematical expressions. Each expression is composed of candidate constitutive terms combined using different mathematical operators. Then, the performance of each expression is evaluated using a fitness function, and the expression with better performance is selected as the parent. A crossover operation is used to exchange and combine the subtrees of two parent expressions to generate new child expressions. A mutation operation is used to randomly change certain nodes or subtree structures in the expressions. After multiple generations of evolution, the optimal expression structure set is finally obtained.

[0040] S32: By using the differential optimization algorithm, the inner layer parameters of multiple formulas in the candidate formula structure set are optimized to obtain the optimal parameter configuration of multiple formulas.

[0041] In step S32, this invention uses a differential optimization algorithm to optimize the inner parameters of multiple formulas in the candidate formula structure set to obtain the optimal parameter configuration of multiple formulas. The aim is to search for the optimal parameter combination by simulating the biological evolution process. The algorithm maintains a population containing multiple parameter vectors, each parameter vector corresponding to a set of candidate parameter values. In each iteration, the algorithm generates a mutation vector for each target vector. The mutation vector is generated by randomly selecting three different individuals from the population and linearly combining them according to a specific mutation strategy. Then, the mutation vector is crossed with the target vector to generate a trial vector. Finally, the fitness values ​​of the trial vector and the target vector are compared to determine whether to accept the trial vector as the next generation of individuals.

[0042] The differential optimization algorithm in step S32 adopts the best1bin strategy.

[0043] Furthermore, the differential optimization algorithm employs the best1bin strategy, where best1 indicates that the best individual in the current population is used as the basis vector in the mutation operation, and bin indicates that a binomial crossover operation is used. The specific mathematical expression is as follows: ; in It is a mutation vector. It is the most fit individual in the current population. and They are two different individuals randomly selected from the population. F It is the mutation factor, with a value between 0.4 and 1.0, while the binomial crossover operation randomly selects each dimension of the mutated vector and the target vector according to the crossover probability CR.

[0044] S33: The optimal parameter configuration is evaluated based on the cross-entropy loss function and regularization constraints to obtain the optimal constitutive equation structure and parameter combination.

[0045] Step S33 evaluates the performance of the optimal parameter configuration based on the cross-entropy loss function and regularization constraints to obtain the optimal constitutive equation structure and parameter combination. The cross-entropy loss function is a loss function used to measure the difference between the predicted probability distribution and the true label distribution. The regularization constraint adds a parameter complexity penalty term to the loss function to prevent overfitting. Step S33 of this invention performs a performance evaluation process through the above two rules, that is, by calculating the comprehensive loss value of each candidate formula on the validation set to compare the performance of different formulas, and selecting the formula with the smallest loss value as the optimal constitutive equation structure and parameter combination.

[0046] S4: Verify the interpretability of the optimal constitutive equation structure and the parameter combination, and establish a constitutive model for the engineering problem based on the verified optimal constitutive equation structure and parameter combination.

[0047] Furthermore, the comprehensive evaluation of the optimal constitutive equation structure and parameter combination in step S4 of this invention aims to compare and verify the mathematical expression obtained by symbolic regression with actual physical laws in order to obtain a constitutive model based on the constitutive equation.

[0048] Step S4 further includes: S41: Physical constraint verification is performed to check the physical rationality of the optimal constitutive equation structure and parameter combination, and the physical constraint verification results are obtained.

[0049] Furthermore, the physical constraint verification is achieved through numerical calculation. Specifically, firstly, dimensional analysis is performed on each physical term in the optimal constitutive equation to ensure that the dimensions on both sides of the equation remain consistent. Taking the material constitutive relation as an example, during data processing, the stress value from the experimental data is substituted into the left side of the equation, and the corresponding strain, strain rate, and temperature values ​​are substituted into the terms on the right side. The calculated numerical results are compared with the actual stress values. When the strain increases from 0.001 to 0.01, the calculated stress value must show a monotonically increasing trend. If a negative value or a non-physical abrupt change occurs, then the constitutive equation structure does not satisfy the physical constraint conditions. For example, the physical rationality test of the temperature effect involves analyzing the coefficient of the exponential term. When the temperature increases from room temperature (293K) to high temperature (673K), if the coefficient of the exponential term is positive, the stress increases with the increase of temperature, which is reasonable in the thermal strengthening phenomenon of materials. If the coefficient of the exponential term is negative, the stress decreases with the increase of temperature, which is consistent with the thermal softening law of most metallic materials. In the data processing, the relative error between the predicted stress value and the experimental measurement value at different temperatures is calculated. When the relative error exceeds the set threshold, it is considered that the parameter combination does not meet the physical constraints.

[0050] The final physical constraint verification results are stored in Boolean form, including multiple dimensions such as dimensional consistency test results, monotonicity test results, and boundary condition test results. In the results, the constitutive equation is considered to satisfy the physical rationality requirements only when all physical constraint test terms are true.

[0051] S42: Based on the accuracy index and the verification results of the physical constraints, evaluate the optimal constitutive equation structure and parameter combination to obtain the constitutive model of the engineering problem.

[0052] Furthermore, in the evaluation and processing stage of step S42, the accuracy index and the physical constraint verification results are quantitatively fused. The accuracy index is obtained by calculating the root mean square error (RMSE) between the predicted and experimental values. The physical constraint verification results are quantified in the form of a score. During the evaluation and processing, the final score is calculated using a weighted fusion method. In a specific embodiment, when the comprehensive score exceeds a preset threshold (set to 0.9 in this invention), the optimal constitutive equation structure and parameter combination pass comprehensive verification and are formally established as the constitutive model for the engineering problem.

[0053] S5: Solve the engineering problem to be solved using the constitutive model of the engineering problem to obtain the solution result.

[0054] The engineering problems to be solved in step S5 include: process parameter optimization problem, cross-scale performance prediction problem, and finite element simulation problem.

[0055] Furthermore, the data processing flow for process parameter optimization problems includes using the constitutive model as the core component of the objective function, and calculating the corresponding material performance output by adjusting input variables such as processing temperature and deformation rate; cross-scale performance prediction requires mapping microscale material parameters to macroscale engineering performance through constitutive equations, and scale transformation and parameter transfer are required during data processing; in finite element simulation problems, the constitutive model is the core of material property definition, and the continuous stress-strain relationship needs to be discretized into numerical calculations at finite element nodes.

[0056] like Figure 2 As shown, the present invention also provides an engineering problem solving system based on constitutive model-driven methods, comprising: Mining module 100: Used to mine physical terms from prior knowledge to obtain a set of candidate constitutive terms; Enhancement module 200: Used to augment experimental data using a Gaussian noise function to obtain an expanded training dataset; Optimization module 300: used to perform nested optimization on the candidate constitutive term set to obtain the optimal constitutive equation structure and parameter combination; Verification module 400: used to verify the interpretability of the optimal constitutive equation structure and the parameter combination, and to establish a constitutive model of the engineering problem based on the verified optimal constitutive equation structure and parameter combination; The solution module 500 is configured to solve the engineering problem constitutive model established by the verification module 400 and obtain the solution result.

[0057] The present invention also provides a constitutive model-driven engineering problem-solving device, comprising: A memory and at least one processor, wherein the memory stores instructions; At least one of the processors invokes the instructions in the memory to cause a constitutive model-driven engineering problem-solving apparatus to execute a constitutive model-driven engineering problem-solving method as described in any of the preceding claims.

[0058] The present invention also provides a computer-readable storage medium storing instructions that, when executed by a processor, implement a constitutive model-driven engineering problem-solving method as described in any one of the above.

[0059] Furthermore, the constitutive model-driven engineering problem-solving device provided by this invention can vary considerably due to different configurations or performance. It may include one or more central processing units (CPUs), for example, one or more processors and memory, and one or more storage media for storing applications or data, such as one or more mass storage devices. The memory and storage media may be temporary or persistent storage. The program stored in the storage media may include one or more modules, each module may include a series of instruction operations on the constitutive model-driven engineering problem-solving device. Furthermore, the processor may be configured to communicate with the storage media and execute a series of instruction operations in the storage media on the constitutive model-driven engineering problem-solving device.

[0060] It may also include one or more power supplies, one or more wired or wireless network interfaces, one or more input / output interfaces, and one or more operating systems, such as Windows Server, MacOSX, Unix, Linux, FreeBSD, etc. Those skilled in the art will understand that the constitutive model-driven engineering problem-solving device structure provided by this invention does not constitute a limitation on the device, and may include more or fewer components than illustrated, or combine certain components, or have different component arrangements.

[0061] The following describes a constitutive model-driven method and system for solving engineering problems according to the present invention, with reference to a specific embodiment.

[0062] The final model in this embodiment is a prediction model for the tendency of hot cracking during welding repair of K439B high-temperature alloy.

[0063] K439B is a new type of nickel-based high-temperature alloy casing material with a resistance to 800℃. It features excellent casting performance and high-temperature strength, and is often used for repair welding with K439B body welding wire. Target quantity: Hot cracking index (CPI) of repair welding; variables: welding current, single pass duration, base metal thickness, weld depth, weld area, preheating temperature, interpass temperature, time to cool to below 500℃ after welding, and welding wire diameter, for a total of 9 variables.

[0064] Based on existing experience, the nine variables were transformed into six local physical terms, and the existing 32 data points were expanded to 32,000 through data augmentation techniques to serve as the training set.

[0065] Then, candidate formulas are generated. At the beginning of each iteration, candidate formulas are generated through the FormulaGenerator class.

[0066] The optimization process is as follows: Generation 1 optimization: Initially generate 100 formulas containing all terms from t1 to t6. Example of candidate formulas: ; ; After parameter optimization, the optimal formula is: The accuracy rate was 76%.

[0067] Second-generation optimization: Generates new formulas based on the first-generation elite model. Example of candidate formulas: ; ; After parameter optimization, the optimal formula is: The accuracy rate is 80%.

[0068] Generations 3 through 5: Continue testing different combinations of operators, and finally optimize and confirm the best formula: Final accuracy: 89%, optimal parameters: Thus, a prediction model for the tendency of hot cracking in the repair welding of K439B high-temperature alloy was obtained.

[0069] This invention achieves the co-evolution of structure and parameters of engineering constitutive models through a double-layer nested optimization strategy. The outer layer uses symbolic regression to search for the optimal formula structure, while the inner layer uses differential evolution to optimize parameter configuration. This breaks through the technical bottleneck of traditional machine learning models lacking physical interpretability, effectively solves the problem of data scarcity in the engineering field, significantly improves model training performance, and ensures that the established constitutive model meets both mathematical accuracy requirements and physical laws. It also achieves a balanced optimization of prediction performance and interpretability.

[0070] The established constitutive model for engineering problems can be directly applied to practical engineering scenarios such as process parameter optimization, cross-scale performance prediction, and finite element simulation. It provides intelligent solution schemes for complex engineering problems that combine high accuracy and strong interpretability, and greatly improves the efficiency and reliability of engineering design.

[0071] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the embodiments of the present invention, and are not intended to limit them. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention. Any changes or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in the present invention should be included within the protection scope of the present invention.

Claims

1. A constitutive model-driven method for solving engineering problems, characterized in that, include: S1: Perform physical term mining on prior knowledge to obtain a set of candidate constitutive terms; including: Determine the target quantity of the engineering problem, which includes the hot cracking index of the repair weld; The relevant variable parameters are determined based on the hot cracking index of the repair weld. The relevant variable parameters include welding current, single pass duration, base metal thickness, weld depth, weld area, preheating temperature, interpass temperature, cooling time to below 500°C after welding, and welding wire diameter. Identify the physical relationships between variables and group related variables into local physical terms; By adding exponential functions to multiple local physics terms to eliminate dimensions, a standardized set of candidate constitutive terms is obtained. The candidate constitutive term set includes current thermal input terms, geometric constraint terms, temperature gradient terms, heat accumulation terms, cooling terms, and wire diameter terms; S2: Data augmentation of the experimental data is performed using a Gaussian noise function to obtain an expanded training dataset; S3: Perform nested optimization on the candidate constitutive term set to obtain the optimal constitutive equation structure and parameter combination; Step S3 further includes: S31: Using the symbolic regression algorithm, perform an outer structure search on the candidate constitutive term set to obtain a candidate formula structure set; S32: By using the differential optimization algorithm, the inner layer parameters of multiple formulas in the candidate formula structure set are optimized to obtain the optimal parameter configuration of multiple formulas; S33: The optimal parameter configuration is evaluated based on the cross-entropy loss function and regularization constraints to obtain the optimal constitutive equation structure and parameter combination; S4: Verify the interpretability of the optimal constitutive equation structure and the parameter combination, and establish a constitutive model for the engineering problem based on the verified optimal constitutive equation structure and parameter combination; including: The physical rationality of the optimal constitutive equation structure and parameter combination is verified by physical constraint verification, and the physical constraint verification results are obtained. The accuracy index is quantitatively integrated with the physical constraint verification results, and the final score is calculated through weighted fusion. When the comprehensive score exceeds the preset threshold, the optimal constitutive equation structure and parameter combination are fully verified, and a prediction model for the tendency of hot cracking in high-temperature alloy repair welding is obtained. S5: Solve the engineering problem of solving the hot cracking index of the high-temperature alloy repair welding using the aforementioned high-temperature alloy repair welding hot cracking tendency prediction model, and obtain the solution results.

2. The constitutive model-driven method for solving engineering problems according to claim 1, characterized in that, The set of candidate constitutive terms in step S1 is selected based on the physical mechanism of the engineering domain corresponding to the engineering problem to be solved.

3. The constitutive model-driven method for solving engineering problems according to claim 1, characterized in that, The differential optimization algorithm in step S32 adopts the best1bin strategy.

4. A constitutive model-driven engineering problem-solving system, characterized in that, include: Mining module: Used to mine physical terms from prior knowledge to obtain a set of candidate constitutive terms; Determine the target quantity of the engineering problem, which includes the hot cracking index of the repair weld; The relevant variable parameters are determined based on the hot cracking index of the repair weld. The relevant variable parameters include welding current, single pass duration, base metal thickness, weld depth, weld area, preheating temperature, interpass temperature, cooling time to below 500°C after welding, and welding wire diameter. Identify the physical relationships between variables and group related variables into local physical terms; By adding exponential functions to multiple local physics terms to eliminate dimensions, a standardized set of candidate constitutive terms is obtained. The candidate constitutive term set includes current thermal input terms, geometric constraint terms, temperature gradient terms, heat accumulation terms, cooling terms, and wire diameter terms; Augmentation module: Used to augment experimental data using a Gaussian noise function to obtain an expanded training dataset; Optimization module: used to perform nested optimization on the candidate constitutive term set to obtain the optimal constitutive equation structure and parameter combination; including: By using the symbolic regression algorithm, an outer structure search is performed on the candidate constitutive term set to obtain a candidate formula structure set; By using the differential optimization algorithm, the inner layer parameters of multiple formulas in the candidate formula structure set are optimized to obtain the optimal parameter configuration of multiple formulas. The optimal parameter configuration is evaluated based on the cross-entropy loss function and regularization constraints to obtain the optimal constitutive equation structure and parameter combination. Verification module: used to verify the interpretability of the optimal constitutive equation structure and the parameter combination, and to establish a constitutive model of the engineering problem based on the verified optimal constitutive equation structure and parameter combination; including: The physical rationality of the optimal constitutive equation structure and parameter combination is verified by physical constraint verification, and the physical constraint verification results are obtained. The accuracy index and the physical constraint verification results are quantitatively integrated, and the final score is calculated through a weighted fusion method. When the comprehensive score exceeds the preset threshold, the optimal constitutive equation structure and parameter combination pass the comprehensive verification, and a high-temperature alloy repair welding hot crack tendency prediction model is obtained. The solution module is configured to use the high-temperature alloy repair welding hot crack tendency prediction model established by the verification module to solve the engineering problem of repair welding hot cracking index and obtain the solution result.

5. A constitutive model-driven engineering problem-solving device, characterized in that, include: A memory and at least one processor, wherein the memory stores instructions; At least one of the processors invokes the instructions in the memory to cause a constitutive model-driven engineering problem solving device to execute a constitutive model-driven engineering problem solving method as described in any one of claims 1 to 3.

6. A computer-readable storage medium, characterized in that, The computer-readable storage medium stores instructions that, when executed by a processor, implement a constitutive model-driven engineering problem-solving method as described in any one of claims 1 to 3.

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